Transcript PowerPoint Presentation - Physics 121. Lecture 10.

Physics 121.
Thursday, February 21, 2008.
Conservation of energy!
Changing kinetic energy into thermal energy.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Physics 121.
Thursday, February 21, 2008.
• Topics:
• Course information
• Review of the concept of work and kinetic energy.
• Conservation laws: why do we care?
• Conservative and non-conservative forces.
• Potential energy.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Course information.
Homework assignments.
• Homework set # 4 is due on February 23 at 8.30 am.
• There will be no homework set due on Saturday March 1.
• Homework set # 5 will be available on the WEB on
Thursday morning, February 28, at 8.30 am.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Asking for help. The more details provided,
the better the answer.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Course information.
Exam # 1.
• On Thursday February 28 between 8 am and 9.30 am the
first midterm exam of Physics 121 will be held. The
material covered on the exam is the material covered in
Chapters 1 - 6 of our text book.
• The location of the exam is Hubbell auditorium.
• There will be a normal lecture after the exam (at 9.40 am in
Hoyt).
• A few remarks about the exam:
• You will be provided with an equation sheet with all important
equations used in Chapter 1 - 6.
• If you show up late, you will just have less time to complete the
exam.
• If you miss the exam, except for valid well-documented medical
reasons, you will receive a score of 0 points. Having your alarm
clock die overnight is not considered a valid medical reason.
• Any makeup exam will be a 90-minute oral exam.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Course information.
Exam # 1.
• During workshops on Monday 2/25, Tuesday 2/26, and
Wednesday 2/27, the focus will be exam # 1. You can attend
any (or all) workshops on these days. Bring your questions!
• There will be no workshops and office hours on Thursday
2/28 and Friday 2/29.
• There will be extra office hours on Wednesday 2/27.
• A Q&A session on the material covered on exam # 1 will
take place on Tuesday evening 2/26. Time and place will be
announced via email.
• You will receive the exam back during workshop during the
week of March 3.
• The TAs will not see the exam until you see it.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Physics 121.
Quiz lecture 10 (postponed).
• The quiz today will have 3 questions.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
A quick review.
Work and energy.
• When a force F is applied to an
object, it may produce a
displacement d.
• The work W done by the force F
is defined as
r r
W  Fgd  Fd cos
where  is the angle between the
force F and the displacement d.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
A quick review.
Work: positive, zero, or negative.
• Work done by a force can be
positive, zero, or negative,
depending on the angle :
• If 0° ≤  < 90° (scalar product
between F and d > 0) the speed of
the object will increase.
•  = 90° (scalar product between F
and d = 0) the speed of the object
will not change.
• If 90° <  ≤ 180° (scalar product
between F and d < 0) the speed of
the object will decrease.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
A quick review.
The work-energy theorem.
• There is a connection between the work done by a force and the change
in the speed of the object:
• If W > 0 J: speed increases
• If W = 0 J: speed remains constant
• If W < 0 J: speed decreases
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
A quick review.
The work-energy theorem.
• The work-energy theorem:
The net work done on an object is equal to the change in its kinetic energy.
The kinetic energy K of an object with mass m, moving with velocity v, is
equal to (1/2)mv2.
• In the case of the bus shown above: Fnet d = 0.5mv22 - 0.5mv12.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Conservation of energy.
• Conservation laws in physics can be expressed in the
following manner:
“Consider a system of particles, completely isolated from outside
influence. As the particles move about and interact with each other,
there are certain properties of the system that do not change.“
• One of the properties of closed systems that will not change
is the total energy of the system. The energy may be
converted from one form to another form, but the total will
not change. Note: you never waste energy; you just
transform it from a useful form to a useless form when you
waste it!
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Conservation of energy.
• Before we can apply conservation
of energy, we need to define what
the total energy of the system is.
• Clearly, the total energy is not
equal to the kinetic energy of the
components since this sum is
clearly not conserved.
• In addition to kinetic energy, the
system must contain potential
energy (energy that has the
potential to be converted into
kinetic energy) in order to be
conserved.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Conservation of energy.
• Consider what would happen if we define the mechanical
energy of a system to be equal to the sum of the kinetic
energy K and the potential energy U:
E=K+U
• If the total mechanical energy is constant, we must require
that DE = 0, or
DK + DU = 0
• We conclude any change in the kinetic energy DK must be
accompanied by an equal but opposite change in the
potential energy DU.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy in one dimension.
• Per definition, the change in potential energy is related to
the work done by the force:
x
DU  W    F x 'dx '
x0
• The potential energy at x can thus be related to the potential
energy at a point x0:
x
U x   U x0  DU  U x0   F x 'dx '
x0
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy in one dimension.
x
U x   U x0  DU  U x0   F x 'dx '
x0
• When we apply conservation of energy, we are in general
only concerned with changes in the potential energy, DU,
and not the actual value of U.
• We are free to assign a value of 0 J to the potential energy
when the system is in its reference configuration.
• Note: the units of potential energy are the units of energy
(the Joule).
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy.
Path dependence.
• The difference between the
potential energy at (2) and at (1)
depends on the work done by the
force F along the path between
(1) and (2).
• But ……. we can get from (1) to
(2) via path A and via path B. In
order to uniquely define the
potential at (2) the work done
must only depend on the start and
end point, and not on the path
followed.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy.
Path dependence.
• The work done must only depend
on the start and end point, and not
on the path followed.
• This is not true for all forces. For
example, the work done by the
friction force is always negative.
If the friction force is constant in
magnitude, the work done by the
friction force depends on the path
length and is thus path dependent.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy.
Conservative and non-conservative forces.
• If the work is independent of the
path, the work around a closed
path will be equal to 0 J.
• A force for which the work is
independent of the path is called
a conservative force.
• A force for which the work
depends on the path is called a
non-conservative force.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy in one dimension.
• The potential energy is directly
related to the force acting on the
object.
F=0N
• If we know the force, we can
calculate the change dU:
x
DU  W    F x 'dx '
x0
• If we know the change dU, we
can calculate the force:
dU
F x   
dx
Frank L. H. Wolfs
Stable
Equil.
F<0N
F>0N
Department of Physics and Astronomy, University of Rochester
Potential energy.
Examples: the spring force.
• The force exerted by a spring is
given by Hooke’s Law:
F x   kx
• Consider the potential energy at
x, taking the rest position as our
reference point (where U = 0 J):
x
U x   U x0   F x 'dx ' 
x0
x
1 2
   kx 'dx '  kx
2
0
• We see that the potential energy
has a minimum when x = 0 m.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Potential energy.
Examples: the gravitational force.
• Consider what happens when we
increase the height of an object of
mass m by h.
• The work done by the
gravitational force is negative
(force and displacement point in
opposite directions) and equal in
magnitude to mgh.
• The potential energy due to the
gravitational force at position 2 is
thus equal to
U y2   U y1  W  U y1   mgh
Usually the potential energy on
the surface is taken to be 0 J.
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Conservation of Mechanical Energy
• When we drop an object, the
potential energy of the object is
converted to kinetic energy.
• At a height h, the total
mechanical energy is mgh.
• At ground level, the total
mechanical energy is (1/2)mv2.
• If
mechanical
energy
is
conserved we conclude that
(1/2)mv2 = mgh
or
v = (2gh)1/2
which is independent of m.
Frank L. H. Wolfs
Note: you could also use the
equations of motion to find v.
Department of Physics and Astronomy, University of Rochester
Conservation of mechanical energy.
• In this example we convert
potential energy to mechanical
energy and back to potential
energy.
• The initial mechanical energy is
all in the form of gravitational
potential energy and is equal to
mgh.
• The final mechanical energy is
the sum of the gravitational
potential energy and the potential
energy associated with the
compressed spring and is equal to
(1/2)kY 2 - mgY.
Frank L. H. Wolfs
U=0J
Department of Physics and Astronomy, University of Rochester
Conservation of mechanical energy.
U = mgh
K=0J
U = mgh
K=0J
U=0J
K = (1/2)mv2
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
Mechanical energy and work.
• Let’s test our understanding of the concepts of mechanical
energy and work by working on the following concept
problems:
• Q10.1
• Q10.2
• Q10.3
• Q10.4
• Q10.5
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester
That’s all!
Next week: more on conservation of energy.
Unusual Spherules on Mars
Credit: Mars Exploration Rover Mission, JPL, USGS, NASA
Frank L. H. Wolfs
Department of Physics and Astronomy, University of Rochester